Polycon

In geometry, a polycon is a kind of a developable roller.

[1][2] In principle, there are infinitely many polycons, as many as there are even sided regular polygons.

of which coincide with the polygon's vertices, with the remaining two lying at the extreme ends of the solid.

edges, each one being half of the conic section created where the cone's surface intersects one of the two cutting planes.

edges of the polycon run (from every second vertex of the polygon) to one of the solid's extreme ends.

As a poly-sphericon, it is constructed by cutting a bicone with an apex angle of

at its plane of symmetry and reuniting the two obtained parts after rotating them at an offset angel of

[5] As a polycon, the starting point is a cone created by rotating two adjacent edges of a square around its axis of symmetry that passes through their common vertex.

In this specific case there is no need to extend the edges because their ends reach the square's other axis of symmetry.

Since, in this specific case, the two cutting planes coincide with the plane of the cone's base, nothing is discarded and the cone remains intact.

The additional vertices are not noticeable because they are located in the middle of the circular edges, and merge with them completely.

[1] The surface of each polycon is a single developable face.

Each of the two extreme vertices touches the rolling plane, instantaneously,

The instantaneous line of contact between the polycon and the surface it is rolling on is a segment of one of the generatinglines of a cone, and everywhere along this line the tangent plane to the polycon is the same.

is an odd number this tangent plane is a constant distance from the tangent plane to the generating line on the polycon surface which is instantaneously uppermost.

odd, are constant height rollers[citation needed] (as is a right circular bicone, a cylinder or a prism with Reuleaux triangle cross-section).

[1] The sphericon was first[dubious – discuss] introduced by David Hirsch in 1980[6] in a patent he named 'A Device for Generating a Meander Motion'.

Only more than 25 years later, following Ian Stewart's article about the sphericon in the Scientific American Journal, it was realized both by members of the woodturning [17, 26] and mathematical [16, 20] communities that the same construction method could be generalized to a series of axial-symmetric objects that have regular polygon cross sections other than the square.

The new family was first introduced at the 2019 Bridges Conference in Linz, Austria, both at the art works gallery[6] and at the film festival[8]